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In order to motivate examples in the first class in congruence theory, my teacher remarked that the beginning chapters of the Holy Bible mathematically said entail the following: "Let the days of the week be congruent modulo seven."
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UnknownJun 12 '10 at 17:07

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Why did a question with so much positive feedback get closed?
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RomeoNov 28 '10 at 23:21

@Matt: standards for what kind of questions people want on MO have changed over time, and keeping this question opens gives a false impression to new users of what kind of questions we want on MO. It's less confusing to close it. This happens on other SE sites as well; many of the most popular questions on StackOverflow, for example, are also closed. There's also the more practical issue that if it's open people keep adding answers and, again, the marginal utility of each additional answer is decreasing.
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Qiaochu YuanNov 12 '13 at 3:03

"Maybe at times I like to give the impression, to myself and hence to others, that I am the easy learner of things of life, wholly relaxed, "cool" and all that - just keen for learning, for eating the meal and welcome smilingly whatever comes with it's message, frustration and sorrow and destructiveness and the softer dishes alike. This of course is just humbug, an images d'Epinal which at whiles I'll kid myself into believing I am like. Truth is that I am a hard learner, maybe as hard and reluctant as anyone."

At the risk of overloading an already bloated thread, I found a rather large collection here. Example:

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

@Jonas: nice link! I especially enjoyed Korner's remark later on: "It is frequently claimed that Lebesgue integration is as easy to teach as Riemann integration. This is probably true, but I have yet to be convinced that it is as easy to learn."
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Thierry ZellNov 28 '10 at 6:27

Jean Bourgain, in response to the question, "Have you ever proved a theorem that you did not know was true until you made a computation?" Answer: "No, but nevertheless it is important to do the computation because sometimes you find out that more is there than you realized."

Like many people, I am fascinated by the quote from Weyl (already listed
here), that

In these days the angel of topology and the devil of abstract algebra
fight for the soul of each individual mathematical domain.

But I can see why people are puzzled by the quote, so I'd like to add some
more information (too much to put in a comment) as another answer.

First, what is the context? The quote occurs in Weyl's paper Invariants
in Duke Math. J. 5 (1939), pp. 489--502, the first page of which can be seen
here. This page includes most of what Weyl has to say on algebra v.
geometry, though the quote itself does not occur until p.500. Then on p.501
Weyl explains his discomfort with algebra as follows

In my youth I was almost exclusively active in the field of analysis;
the differential equations and expansions of mathematical physics were
the mathematical things with which I was on the most intimate footing.
I have never succeeded in completely assimilating the abstract
algebraic way of reasoning, and constantly feel the necessity of translating
each step into a more concrete analytic form.

Second, why the image of angel and devil? According to V.I Arnold,
writing here, Weyl had a particular image in mind, namely, the
Uccello painting "Miracle of the Profaned Host, Episode 6", which can be
viewed here.

Arnold describes this painting as "representing an event that happened in
Paris in 1290." "Legend" is probably a better word than "event," but in
any case it is a very strange origin for a famous mathematical quote.

"Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether."

Apart from the most elementary mathematics, like arithmetic or high school algebra, the symbols, formulas and words of mathematics have no meaning at all. The entire structure of pure mathematics is a monstrous swindle, simply a game, a reckless prank.
You may well ask: "Are there no renegades to reveal the truth?"
Yes, of course. But the facts are so incredible that no one takes them seriously. So the secret is in no danger. -- T. Kaczynski

Another quote from Dieudonné's "Foundations of Modern Analysis, Vol. 1":

The reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the "Riemann integral". It may well be suspected that, had it not been for its prestiguous name, this would have been dropped long ago, for (with due respect to Riemann's genius) it is certainly quite clear for any working mathematician that nowadays such a "theory" has at best the importance of a mildly interesting exercise [...]. Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance.

`The human is just a creature for doing slower (and unreliably) (a small part of) what we already know (or soon will know) to do faster. All pretensions of human superiority should be withdrawn if humans want to survive in the future.

This is definitely misattributed. Ekhad might be good at proving theorems in combinatorics but I don't think he's quite sentient enough to come up with something like this. [he's a computer]
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Kevin BuzzardNov 30 '09 at 6:52

I hope that's not a true quote. Graham, Knuth & Patashnik's Concrete Mathematics quotes him as saying in "Technical Education and its Relation to Science and Literature" among other things "Civilization advances by extending the number of important operations which we can perform without thinking about them." Which I like much more.
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MioMar 26 '10 at 3:06

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I remember trying to track this down; this is what I found. Although WZ quote it as being from Concrete Mathematics by Graham, Knuth and Patashnik and attribute it to the authors, in the book it is just a margin comment left by one of the students of the Stanford class. (The book is full of those.) It follows Whitehead's quote ("It is a profoundly erroneous truism [...] Civilization advances by extending the number of important operations which we can perform without thinking about them": www-history.mcs.st-and.ac.uk/Quotations/Whitehead.html) and some student must have "extended" it.
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shreevatsaOct 22 '12 at 4:41

2

Well, one could reasonably say that the sentiment is overblown (as are most aphorisms, almost by definition), but another take on it might be that mathematical understanding is full and ripe when every step, every argument, feels natural and inevitable -- eliminating traces of cleverness which appear as if out of nowhere. Such cleverness being felt as jarring in a way, and indicating that there is something left which hasn't yet been truly and deeply understood.
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Todd Trimble♦Jun 9 '13 at 14:32

"We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things." - Anonymous quote from Bernt Øksendal's "Stochastic Differential Equations".

Free translation: «Life is strange. In fact, in geometry, we do not think in the same way of a complex affine line (for example in the theorem of Ceva in a triangle) and of the field of complex numbers x+iy. When I think about this, imaginary points in geomtry are gray, the real points are black, and the intersection of two conjugate imaginary lines is a black real point. The beautiful umbilical conic is silver, the lines and isotropic cones are mostly pink.»

Reminds me of the synaesthetic experiences of Feynman: "When I see equations, I see the letters in colors – I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students."
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Todd Trimble♦Jun 9 '13 at 14:42

"The case for my life, then, or for that of anyone else who has been a mathematician in the same sense which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them." - G.H. Hardy

"'Imaginary' universes are so much more beautiful than this stupidly constructed 'real' one; and most of the finest products of an applied mathematician's fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts." - G.H. Hardy

Dunno if it's appropriate, but:
"Now, I've often thought of writing a mathematics textbook someday, because I have a title that I know will sell a million copies. I'm going to call it: Tropic of Calculus" -- Tom Lehrer, New Math

As many people say it is a example of false modesty, it is a fact that Einstein was poor mathematician. And physics of his times do not require very abstract knowledge. But it was very deep thinker, and very consequent one.
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kakazFeb 28 '10 at 19:30

Explanation given by Newton to Leibniz in response to the latter's request for details about Newton's newly developed method of fluxions and fluents, in the form of an anagram for «Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire, et vice versa».

Who's not shared the feeling that Leibniz must have felt at getting this response when reading obscure explanations in the literature? :P

Translation as given in Additive number theory: inverse problems and the geometry of sumsets, vol. 2, by M. B. Nathanson: «It is true that Fourier believed that the principal goal of mathematics is the public welfare and the understanding of nature, but as a philosopher he should have understood that the only goal of science is the honor of the human spirit, and, in this regard, a problem in number theory is as important as a problem in physics.» The translation sadly loses much of the tone...